Mat210 Section 4.7 - Infinite Intervals of Integration

No doubt you are already an "old hand" at evaluating integrals. The fundamental theorem of calculus gave us this simple and amazing result: the area under a curve is equal to the antiderivative evaluated at each endpoint. An example to remind you of this process (not that you need it!) is given below.

The upshot is, no matter which of the family of antiderivatives one chooses, the number obtained for the area under the derivative curve over the specified interval will always be F(b) - F(a).

But what if the interval is infinitely long? That is, what if you had something like ? Would this even make sense? In the case of the function y = 2x, the area under that line over the interval from zero to infinity would also be infinitely large, so this is not so useful or informative (we say the integral diverges, that is, doesn't add up to any number) . But there are instances where (believe it or not!) you do end up with a finite, real number, even though you have integrated under a curve that is infinitely long. In that case we say that the integral converges. How can that be??

Just to give you a simple analogy, think of the following problem. We want to add up an infinitely long list of numbers, as follows:

as the terms continue to add indefinitely. Since we are adding an infinite number of numbers, do you suppose we end up with an infinitely large result, or does this series actually add up to a fixed real number? Think about this for a minute, and then roll your mouse over the series below to check your answer.


Example 1: An integral where one or more of the limits of integration is infinite is said to be an improper integral. The following are examples of improper integrals...

Since infinity is not a number, we can't just find the antiderivative and "plug in infinity", we work around the problem by rewriting a so-called improper integral with an upper or lower limit of a number "n" , and then when we get the result, we take the limit as .
 

Find the limit of the sum of the total area under the curve : 

on the interval from x = 1 to x = . 

Example 2: Can you be sure that using graphs and a little intuition will always work in determining whether or not an improper integral converges? That is, the area under the curve will add up to a real number, and not increase indefinitely? Not necessarily! Consider a very similar function, .

Problem: does this integral converge?
 

Here we see the graph of . It certainly looks like it ought to converge. But it doesn't! So how do we know that, and in general, how can we know for sure if an integral converges or not?  

 
 

DOES NOT CONVERGE!!! 

 This is a famous counter example.  

In evaluating the integral, we see that we get the natural log of an infinitely large value, which is also infinitely large! Don't let this one fool you when you encounter it!

Example 3: Proving that an improper integral converges.

As mentioned in your text, if you can demonstrate that you can find a function that is larger in its absolute value over some infinite integral that converges, then the smaller one must converge.
 

Prove that  converges.  Consider the fact that  

Since we showed that the function  converged in example 1, g(x) must converge as well. 

 

Example 4 : An Application: Miss Simone Patoodie estimates that the rate of increase in the sale of widgets for her new Bigger and Better Widget Company is given by the function , given in thousands of widgets. If this rate continues indefinitely, what will be her total sales in widgets? To obtain the answer, evaluate the improper integral:.

   

Sales will eventually amount to a total of 316,680 widgets. 
Online Homework Section 4.7 Hard Copy   

 

Last Update: July 20, 2008 la copyright 2007 (c) Sharon Walker and theDepartment of Mathematics and Statistics at ASU - all rights reserved